Real Harmonic Analysis

for Geometric Automorphic Forms

Takayuki Oda

Introduction

This is a survey of the results in the last decade on spherical functions overreal semisimple Lie groups, and their applications to the theory of automorphicforms of many variables, mainly due to the author and his collaborators, Theorigin of our project is the former article [40] of the two decades ago by theauthor. So this article might be also considered as its continuation.

It is difficult to have substancial and general results in the theory of au-tomorphic forms of many variables. Therefore our results in this article areclassified into two categories: when the representations generated by automor-phic forms are (a) either small represenations on rather general semisimple Liegroups, (b) or rather general represenations on small Lie groups. The large-ness of representations is measured by the Gelfand-Kirillov dimension, and thelargeness of groups by the R-split rank. The aspect (a) is shown in the subsec-tion 1.5, which discuss the vanishing of the middle Hodge component of certainmodular symbol, and in the subsection 6.2 to discuss the Green functions asso-ciated with modular algebraic cycles. These are also the geometric aspect in ourresults. The other aspect (b) is shown in the section 5 to discuss the generalizedspherical functions on the ”small group” Sp(2,R) related to the various Fourierexpansions of automorphic forms. The remaining part is for the preparation togive the backgrounds for these results.

But before to proceed to the outline of this paper, let me recall some personalmemory about Sp(2,R) and its representations.

Sometime around 1980, in the Hongo Campus of the University of Tokyoon a slope west of the Yasuda Hall, Ihara who had been my former teacherasked my opinion about the non-holomorphic discrete series representations ofSp(2,R): ”What should be the role of such representations in the arithmeticof automorphic forms on Sp(2,R)?”. A part of the reason of this questionwas because he noticed a relation between the characters of finite-dimensionalrepresentations of the compact form Sp(2) of Sp(2,R) and the characters of thediscrete series representations of Sp(2,R), which became available by a result ofHirai [8] at that time. Also he investigated automorphic forms on the compactgroup Sp(2) (cf. [16]) sometime ago.

The specific problem, which might probably be in his mind, was developedby Ibukiyama [13] and Hashimoto-Ibukiyama [14] (or by Ibukiyama-Ihara [15]?), but the general question has been lingered in my mind. This was one of themotivations for me to write [40].

For the lack of the knowledge of real harmonic analysis on real semisimple Lie

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groups, I have no much progress for long time. But around early 90’s, I noticedthe series of papers by Yamashita ([52] and related papers). This gives me themethod to compute various spherical functions on Sp(2,R) and on SU(2, 2).And fortunately I had many bright students to work out. These are the resultspresented in §5 basically.

Meanwhile there were development in cohomological representations in 80’sand 90’s. Obviously these are related to the modular symbols on higher di-mensional modular varieties, which have been my interest from the time of thedoctoral thesis on Hilbert modular surfaces. Modular symbols on the higherdimensional arithmetic quotients are the theme of the subsections 1.5 and 6.2mentioned above.

Let me explain the contents of this paper. The first section is mainly devotedto a short review of the ”classical” results on the cohomology groups of the(cocompact) discrete subgroups Γ of real semisimple Lie groups G (§§1.1 ∼1.4, 1.6). We recall here the Matsushima isomorphism and the relative Liealgebra cohomology. The exception is the subsection 1.5, here I review shortlythe content of the joint paper with Toshiyuki Kobayashi [23]. The subsection1.6 contains the important example, i.e., the case G = Sp(2,R). The section2 is for non-cocompact Γ in G = Sp(2,R). Here we ”review” the analogue ofEichler-Shimura isomorphism for the third cohomology of Γ, which is a variantof the Matsushima isomorphism, in the context of (g,K)-modules (Theorem2.1). Though we do not claim the result here is new, but the statement relatedto the Dirac-Schmid operator does not seem to be found in the literature. In§3, we recall the ”classical” results by Harish-Chandra on the discrete seriesrepresentations, but mainly limited to the case G = Sp(2,R) (§§3.1∼ 3.3). Andwe also recall the important results of W. Schmid on the realization of thediscrete series and the characterization of the minimal K-types (the sections3.5 and 3.4, respectively), which are the deep and crucial results in the discreteseries. The section 4 is the preparation for the next section 5. We review thegeneral frame work of the Fourier expansion of automorphic forms with respectto relatively ”large” closed subgroup R, and the related problem of sphericalfunctions (the subsections 4.1 and 4.2). The section 5 is the review of the variousspherical functions related to the Fourier expansions of automorphic forms withrespect to the different type parabolic subgroups. In the subsection 6.1, wecomment about spherical functions with respect to symmetric pairs (G, R). Andin the subsection 6.2, we see how the spherical functions are utlized to have aglobal object in the theory of automorphic forms taking the Green function asan example.

The reference list of this paper is already rather long. I am forced to skipeven the papers which are historically important, if they are not directly relatedto this article. They are probably found in the references in the quoted papers.

The contents of this article grew in the course of a few occations to talk atsome workshops. I had a chance to give a series of talks at Tsuda College inSeptember 2000 under the title ”Geometry of arithmetic quotients of the classi-cal domains”. Also the English version of this talk is given at a Japan-Germanyseminar at 2001 under the title ”Modular cycles on arithmetic quotients of clas-sical domains”. These correpond to the section 1 and the subsection 6.1 of thisarticle. The contents of §5 is presented at the workshop: ”Motifs and cyclesof abelian varieties”, January 2004 at Tohoku University, under the title ”Theknown and unknown on Fourier expansions of automorphic forms”. The author

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thanks to the organizer of these workshops. He also thanks to the former stu-dents who obtained excellent results spending much time of concentration fordifficult computaions. Finally he thanks to the organizers of the this workshopat Texel Island, for giving me the chance to write this.

1 The Matsushima isomorphism for cohomol-ogy of discrete subgroups

In this section we recall some basic facts around the Matsushima isommorphismon cohomology groups of discrete subgroups in real semisimple Lie groups.

A good reference on this theme is a survey artilce by A. Borel [1]. The book[2] of Borel-Wallach also has been a very important reference. The original paperabout this theme is Matsushima’s [27]. In the case when G/K is Hermitian,there is a paper by the author which is more specialized to the Hodge theoreticaspect of the problem [40].

1.1 Shift to the relative Lie algebra cohomology groups

We start with a given cocompact discrete subgroup Γ in a real semisimple Liegroup G with finite center. For simplicity we may asuume that G is connected.We consider its Eilenberg-Maclane cohomology group Hi(Γ,C). Or more gener-ally, if a finite dimensional rational representation r : G → GL(E) of G is given,we may regard it as Γ-module by restriction, and can form cohomology groupHi(Γ, E).

Fix a maximal compact subgroup K of G to get a Riemannnian symmetricspace X = G/K. For simplicity assume that we have no elements of finite orderin Γ, then Γ acts on X from the left side without fixed point, and the quotientΓ\X becomes a manifold. In this case Γ is isomorphic to the fundamental groupof this quotient manifold (X is contractible to a point), and the Γ-modules Edefines a local system E on this quotient manifold. Then we have a canonicalisomorphism of cohomology groups: H∗(Γ, E) ∼= H∗(Γ\X, E).

Let σ : X → Γ\X be the canonical map, then by pulling back differentialforms with respect to σ we have a monomorphism of de Rham complexes σ0 :Ω∗Γ\X → Ω∗X . Moreover on the target complex, Γ acts naturally. Let (Ω∗X)Γ bethe invariant subcomplex. Then it coincides with the image of σ0, and by thede Rham theorem, we have an isomorphism of cohomology groups:

H∗(Γ, E) = H∗(Γ\X, E) = H∗(ΩX(E)Γ).

The last complex in the above isomorphims is identified with the complex ofdifferentail forms on G as follows.

Let π0 be the pull-back homomorphism of differential forms with respectto the canonical map π : Γ\G → Γ\X. Then a form ω ∈ Ωd

X(E)Γ defines adifferential form ω0 on G by

x ∈ G 7→ r(x)−1π0(ω)(x),

and denote by A∗0(G, Γ, E) the image of this homomorphism. Then since σ0 is amonomorphism, we have an isomorphism of complexes Ω∗X(E)Γ ∼= A∗0(G, Γ, E).

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Here the last complex is identified with the right K-invariant subcomplexof the de Rham complex Ω∗Γ\G(E) = Ω∗Γ\G ⊗E, which is defined over Γ\G andtakes values in E.

Since the tangent space at each point of G/K is identified with the orthogonalcomplement p of k in g with resepect to the Killing form, the module of the i-thcochains becomes HomK(∧ip, C∞(Γ\G) ⊗ E). Therefore, the cochain comlexdefined in this manner gives the relative Lie algebra cohomology groups. WhenG is connected we have an isomorphism:

H∗(Γ, E) ∼= H∗(g,K; C∞(Γ\G), E).

1.2 Matsushima isomorphism

Let G be a connected semisimple real Lie group with finite center.　 Assumethat the discrete subgroup Γ is cocompact, i.e. the quotinet Γ\G is compact.

Let L2(Γ\G) be the space of L2-functions on G with respect the Haar mea-sure on G, on which G acts unitarily by the right action. By assumption thespace C∞(Γ\G) is a subspace of this space.

Proposition 1.1 (Gelfand, Graev, and Piatetski-Shapiro) If Γ is cocompact,we have the direct sum decomposition of the unitary representation L2(Γ\G)into irreducible components:

L2(Γ\G) = ⊕π∈Gm(π,Γ)Mπ,

with finite multipicities m(π, Γ). Here G is the unitary dual of G, i.e.the unitaryequivalence classes of irreducible unitary representations of G, and Mπ denotesthe representation sapce of π. The symbol ⊕ means the Hilbert space direct sum.

By the above proposition, one has a decomposition

C∞(Γ\G) = ⊕π∈Gm(π, Γ)M∞π

of topological linear spaces. Here M∞π is the subspace consisting of C∞-vectors

in the representation space Mπ.

Theorem 1.1 For a finite dimensional rational G-module E over the complexnumber field, we have an isomorphism:

H∗(Γ, E) = ⊕π∈Gm(π, Γ)H∗(g,K,M∞π ⊗ E)

= ⊕π∈GHomG(Mπ, L2(Γ\G)⊗C H∗(g,K, M∞π ⊗ E)

The key point of the proof here is the complete direct sum ⊕ is replaced by asimple algebraic direct sum by passing to the cohomology (cf. Borel [1]).

We can equipp a K-invaraint inner product on E. By using this, we mayregard Hm(g,K, M∞

π ⊗E) as a space of a kind of harmonic forms, i.e. the totalityof cochains vanishing by the Laplace operator. Thus we have the following.

Proposition 1.2 Let (r,E) be an irreducible G-module of finite dimension. Forany π ∈ G we have the following.(i) If χr(C) = χπ(C), there is an isomorphism

HomK(∧mp, Mπ ⊗∞ ⊗E) = Cm(g,K,M∞π ⊗ E) = Hm(g,K, M∞

π ⊗ E).

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Here Cm(g,K, M∞π ⊗ E) denotes the m-th module of the standard complex of

(g,K) cohomology.(ii) If χr(C) 6= χπ(C), there is an isomorphism

Hm(g,K, M∞π ⊗ E) = 0.

Here C denotes the Casimir operator.In particular, when E is the trivial G-module C, the above theorem is no

other than the original formula of Betti numbers of Γ\X by Matsushima [27].

Also there is a variant of this type vanishing theorem shown by D. Wigner.But it is omitted here.

We recall here that the relative Lie algebra cohomology group Hm(g, K,M∞π ⊗

E) is isomorphic to the continuous cohomology group Hmct(G,M∞

π ⊗ E) if G isconnected. This is shown by using differential cohomology and van Est spectralsequence.

1.3 ”Classical” vanishing theorems

A number of vanishing theorems were found in ’60’s: Calabi-Vesentini, Weiletc. In their proof, the same type of computation of ”curvature forms” is done,which is similar to a proof of Kodaira vanishing theorem.

Firstly, Matsushima’s vanishing theorem of the first Betti number of Γ\Xwas also proved by such method ([28]).

This type of vanishing theorem is vastly improved by representation theoreticmethod. Probably the best resut of this category is the following result byZuckerman ([54]) (see also [2], Chapter V, §2－ §3 (p.150–155)).

Theorem 1.2 Let G be a simple real algebraic group, (π,Mπ) a nontrivial ir-reducible unitary representation of G, and (r, E) a finite dimensional represen-tation of G. Then for k < rankRG,

Hkct(G,M∞

π ⊗ E) = 0.Corollary 1.3 Given a cocompact discrete subgroup Γ of G, for k < rankRG,the restriction homomorphism

Hkct(G,E) → Hk(Γ, E)

is an isomorphism.

1.4 Enumeration and construction of unitary cohomolog-ical representations

We shortly review the state of arts on the cohomological representations definedbelow.

Definition 1.1 An irreducible (unitary) representation (π, Mπ) ∈ G is calledcohomological, if there is a finite dimensional G-module F such that

Hi(g, K; M∞π ⊗C F ) 6= 0

for some i ∈ N. The set of equivalent classes of cohomological representationsis denoted by Gcoh.

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Enumeration of such cohomological representations was done for the casetrivial F = C by Kumaresan [25], and for general case by Vogan-Zuckermann[50]. This was originally described by susing the cohomological induction func-tor Aλ(g) first. And later a global realiztion of this Zuckermann module wasobtained by H.-W. Wong [51].

1.5 A new vanishing theorem

Recently T. Kobayashi is developping a theory of branching rule for such coho-mological representations when they are restricted to a large reductive subgroupH of G ([20, 21, 22]). We have an application of this result to show the vanish-ing of certain Hodge components of some modular symbols (cf. Kobayashi-Oda[23]). This is a bit deeper result of vanishing than the classical ones.

1.5.1 Formulation of problem

Firstly we recall the construction of generalized modular symbols associatedwith redutive subgroups of G. Given a double coset space V = Γ\/K, which iscompact. Let ι : H ⊂ G be a reductive subgroup of G, such that(a) H ∩K is maximally compact in H;(b) H ∩ Γ is cocompact discrete subgroup of H.Then we have a natural map of double cosets:

ι : (H ∩ Γ)\H/(H ∩K) → V.

Set d = dimRW = dimRH/(H ∩ K), N = dimRV = dimRG/K. Then thefundamental class [W ] ∈ Hd(W,C) mapped naturally by ι to

ι∗([W ]) ∈ Hd(V,C) ∼=P HN−d(V,C) ∼= HN−d(Γ,C)

∼= ⊕π∈G,χ∞(pi)=χ∞(1)HomG(Hπ, L2(Γ\G))⊗HN−d(g,K;H∞π ).

Here the first isomorphism denoted by P is the Poincare duality. According tothe last decomposition in the above formula, we have a decomposition

P · ι∗([W ]) =∑

π∈G,χ∞(π)=χ∞(1)

M(π)(W ),

where M(π)(W ) denotes the π-th component of the Poincare dual of [W ].Problem Describe M(π)(W ), using the special values of automorphic L-functions etc, · · ·

Dually speaking, it is to consider the restriction map:

Hd(V,C) = ⊕π∈GHomG(Mπ, L2(Γ\G)⊕C Hd(g,K;Mπ).

→ Hd(W,C) = C.

This is done by invetigation of the period integrals∫

Wω for elements ω in the

π-component

Hd(V,C)(π) = HomG(Mπ, L2(Γ\G)⊕C Hd(g,K; Mπ).

of Hd(V,C).

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A nice situation is when the retriction π|H of π ∈ Gcoh is admissble, i.e.(a) π|H decomposes discretely as H-modules;(b) π|H = ⊕σ∈Hm(π, σ)σ has finite multiplicity m(π, σ) < ∞.

For cohomological representations π = Aλ(q), Koybayashi obtained certainsufficient condition for the admissibility of π|H, which is described geometrically.

As an application, under the same condition as the above criterion of admis-sibility, the restriction to

Hd(W,C)×Hd(V,C)(π)

of the natural pairing:

Hd(V,C)×HddR(V,C) → C

vanishes. Here d = dimW .

1.5.2 Example

Let G = SO(2n, 2) × compact factor such that there is an algebraic group Gover Q with G = G(R). Then we may form an algebraic subgroup H of Gsuch that H = H(R) is isomorphic to SO(2n, 1)× compact factor and there is amaximal compact subgroup K of G with H ∩K being maximal compact in H.

Set X = G/K and XH = H/(H∩K). Then XH is a totally real submanifoldof X. This means that for some holomorphic local coordinates (z1, z2, · · · , z2n)at each point of XH , it is defined locally by the equalities

Im(z1) = 0, · · · , Im(z2n) = 0.

Now for a cocompact arithmetic discrete subgroup Γ of G, we can define ageneralized modular symbol:

ι : W = (H ∩ Γ)\XH → V = Γ\X.

In this case the fundamental class [W ] ∈ H2n(W,C) mapped to

H2n(V,C) ∼= H2ndR(V,C) = ⊕p+q=2nHp,q

by ι∗. Here the last isomorphism is the Hodge decomposition. Hence we havethe Hodge decomposition of M(W ):

M(W ) =∑

p+q=2n

M(p,q)(W ).

Its (n, n)-type component M(n,n)(W ) has further decoposition:

M(n,n)(W ) =∑

π∈G,Hn,n(g,K;H∞π )6=0

M(π)(W ).

Here Hn,n(g,K; M∞π ) is the (n, n)-type component of H2n(g,K; M∞

π ). In thiscase we have the following.

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Proposition 1.3 If π 6= C, M(π))(W ) = 0 for π satisfying Hn,n(g,K; M∞π ) 6=

0. This means that M(n, n)(W ) is a consatnt multiple of a universal coho-mology class η ∈ H(n,n)

dR (V,C), which is one of two natural generators of

Hn,n(g,K;C) = C ∧n κ⊕Cη,

where κ is the Kaehler class. Moreover the constant is vol(W )/vol(V ).

Remark In place of the pair (SO(2n, 2), SO(2n, 1)), we can consider the pair(SU(2n, 2), Sp(n, 1)) for example.

1.6 The Matsushima isomorphism in the case of Hermi-tian G/K

Recall the part (i) of Proposition 1.2:”If χr(C) = χπ(C), there is an isomorphism

Hm(g,K,M∞π ⊗ E) = Cm(g, K, M∞

π ⊗ E) = HomK(∧mpC,M∞π ⊗ E).”

When the quotient X = G/K is Hermitian, the tangent space p at the identity[e] = eK ∈ G/K has the canonical complex struture to give the associateddecomposition: pC = p+ ⊕ p−. Then we have the induced decomposition ofK-module:

∧mpC = ⊕a+b=m ∧a p+ ⊗ ∧bp−.

Thus the last space of the above isomorphism is a direct sum:

HomK(∧mpC,M∞π ⊗ E) ∼= ⊕a+b=mHomK(∧ap+ ⊗ ∧bp−, M∞

π ⊗ E).

1.6.1 Example: the case G = SL(2,R)

Let E = Symk be the symmetric tensor represenation of degree k of the standardrepresentation of G = SL(2,R). It is the unique irreducible finite dimensionalrational representation of G of degree k+1. For a cocompact Γ the Matsushimaisdomorphism is the Eichler isomorphism:

H1(Γ\G/SO(2),Symk) = ⊕π∈G,χ(Symk)∗=χπHomSO(2)(p,Mπ ⊗ Symk)⊗ IΓ(π)

= HomSO(2)(p, MD+k+2

⊗ Symk)⊗ IΓ(D+k+2)⊕

HomSO(2)(p,MD−k+2

⊗ Symk)⊗ IΓ(D−k+2)

= HomG(MD+k+2

, L2(Γ\G))⊕HomG(MD−k+2

, L2(Γ\G))

= Mk+2(Γ)⊕Mk+2(Γ).

Here

HomSO(2)(p,MD+k+2

⊗ Symk) ∼= C and HomSO(2)(p,MD−k+2

⊗ Symk) ∼= C,

and Mk+2(Γ) the space of holomorphic automorphic forms of weight k+2 on theupper half plane SL(2,R)/SO(2) and Mk+2(Γ) the space of antiholomorphicautomorphic forms of weight k + 2.

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1.6.2 The case G = Sp(2,R)

We put

G = Sp(2,R) :=(

A BC D

)= g | tg

(0 12

−12 0

)g =

(0 12

−12 0

) .

Let (ρ(l1,l2), E(l1,l2)) be the irreducible finite-dimensional representation of G

with highest weight l = (l1, l2). Let El be the constructible sheaf on Γ\G/Kassociated with the restriction of El to a cocompact discrete subgroup Γ of G.

There are exactly 4 representations π ∈ G contributing to the non-vanishingcohomology group

H3(g,K; Mπ,K ⊗C El) 6= 0.

Let ρ = 12

∑α>0 α = (2, 1) be the half-sum of positive roots

2e1 = (2, 0), e1 + e2 = (1, 1), 2e2 = (0, 2), e1 − e2 = (1,−1).

Then these 4 discrete series representations of G have Harish-Chandra param-eters (cf. §3 for definition):

(l1 + 2, l2 + 1) = l + ρ, (l1 + 2,−(l2 + 1)) = w1(l + ρ),(l2 + 1,−(l1 + 2)) = w2(l + ρ), (−(l2 + 1),−(l1 + 2)) = w3(l + ρ),

where wi (i = 1, 2, 3) are elements of the Weyl group of Sp(2,R). We denotethem by

π++l+ρ, π+−

w1(+ρ), π−+w2(l+ρ)), π−−w3(l+ρ)),

though the superscripts ±,± are redundant symbols. We have

HomK(∧3p+,Mπ++l+ρ

⊗ El) ∼= C

and

HomK(∧ap+ ⊗ ∧bp−,Mπ++l+ρ

⊗ El) = 0

for the remaining (a, b). We may refer to this fact by saying that π++l+ρ has type

(3, 0). Similarly the remaining representations π+−w1(l+ρ), π−+

w2(l+ρ) and π−−w3(l+ρ)

have the types (2, 1), (1, 2), and (0, 3) respectively. Thus the Matsushima iso-morphism gives an isomorphism:

H3(g,K; Mπ,K ⊗C El) = IΓ(π++l+ρ)⊕ IΓ(π+−

w1(l+ρ))⊕ IΓ(π−+w2(l+ρ))⊕ IΓ(π−−w3(l+ρ)).

For a dominant integral weight λ = (λ1, λ2) of the fixed compact Cartan sub-group T of K ∼= U(2), let (τλ,Wλ) be the associated finite-dimensional irre-ducible representation of K.

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1.6.3 The evaluation map

For example, when we consider π++l+ρ, the non-zero element of HomK(∧3p+,Mπ++

l+ρ⊗

El|K) is realized as follows.Choose the minimal K-type (τmin,Wτmin

) of (π++l+ρ, Mπ++

l+ρ). Then there is

the unique non-zero K-homomorphism

ι : ∧3p+ → τmin ⊗ El|Kwhich is equivalent to give a non-zero element in the K-invvariant space

τmin ⊗ El|K ⊗ ∧3p∗+In our case we have (τmin,Wτmin

) ∼= (τ(l1+3,l2+3,l2+3),W(l1+3,l2+3)). On theother hand we have the unique K-homomorphism i : (τmin,Wτmin

) → (π++l+ρ,Mπ++

l+ρ).

The composition of ι with Hom(id∧3p+ , i⊗idEl) gives the generator of the space

HomK(∧3p+, Mπ++l+ρ

⊗ El|K). On the other hand, for

IΓ(π++l+ρ) = Hom(g,K)(π++

l+ρ, L2(Γ\G)),

we can conider the evaluation map:

Hom(g,K)(π++l+ρ, L

2(Γ\G))⊗Mπ++l+ρ

→ L2(Γ\G),

which is a G-homomorphism.On one part we consider the restriction this homomorphim to the image of

i : τmin → Hπ++l+ρ

, whcih is identified with an element in

f : Γ\G → W ∗τmin

, f(xk) = τ∗min(k)f(x), Xf = 0,∀X ∈ p−,

because π++l+ρ is a highest weight module and the elements in Im(i) is annihilated

by p−. This is the space Ml+ρ(Γ) of vector-valued holomorphic Siegl modularforms of weight (l1 + 3, l2 + 3) belonging to Γ.

Remark 1. On the hand the images of the evaluation map are given by elementsin the space A3(Γ\G/K, El) of C∞ 3-forms with values in El, which is identifiedwith C∞(Γ\G)⊗ El|K ⊗ ∧p∗CK .

We can consider the remaining 3 cases similarly to have analogous spaces:

M+−w1(l+ρ)−ρ(Γ) := f : Γ\G → W(l2+1,−l1−3), C

∞ − function |(i) f(xk) = τ(l2+1,−l1−3)(k)f(x), ∀x ∈ G,∀k ∈ K(ii) D+−f = 0, Here D+− is the Dirac-Schmid operator

M−+w2(l+ρ)−ρ(Γ) := f : Γ\G → W(l1+3,−l2−1), C

∞ − function |(i) f(xk) = τ(l1+3,−l2−1)(k)f(x), ∀x ∈ G,∀k ∈ K(ii) D−+f = 0, Here D−+ is the Dirac-Schmid operator

M−−w3(l+ρ)−ρ(Γ) := f : Γ\G → W(l1+3,l2+3), C

∞ − function |(i) f(xk) = τ(l1+3,l2+3)(k)f(x),∀x ∈ G, ∀k ∈ K(ii) D−−f = 0, Here D−− is the Dirac-Schmid operator.

The precise definition of the Dirac-Schmid operator is not given here. We shalldefine that in later subsections 3.4 and 3.5 to discuss the realization of thediscrete series representations of Sp(2,R).

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2 The case of non-cocompact Γ

We again consider the general G. When Γ is not cocompact, it becomes difficultto desribe the cohomology groups Hi(Γ, E) in terms of automorphic forms. Be-cause the space C∞(Γ\G) contains those functions which increase quite rapidly”at infinity”, i.e., when they approach to the cusps, it is quite inconvenientto develop natural analysis. So we have to impose some conditions at infinity.There are two standard ways to do so.

2.1 Automorphic forms

Before to discuss the Fourier expansion of automorphic forms, we recall theirdefinition. Firstly we have to define the norm

‖g‖ := tr(g · tg).

of an element g ∈ G = Sp(2,R). From now on we let Γ = Sp(2,Z) for simplicity.

Definition An element f ∈ C∞(Γ\G) is han automorphic form, if it satifies thefollowing three conditions:

(i) f is (right) K-finite, i.e., the linear span of right translations Rk(f)(x) =f(xk) (k ∈ K) of f(x) of f by K is of finite dimension;

(ii) f is Z(g)-finite, i.e., there is an ideal I of finite codimension in the centerZ(g) of the univeral enveloping algebra U(g), which annihilate f ;

(iii) There are a natural number N and a positive constant C such that

|f(x)| ≤ C‖x‖N for any x ∈ G.

We denote by A(Γ\G) the space of automorphic forms defined above. It is aG-module by the right action of G which is smooth under an adequate topology,and a (g,K)-module algebraically. If necessary, it is better to replace the secondcondition of the above definition by a more restrictive condition:

(ii)’ f generates an irreducible (g,K)-module Vf by right translation.Remark. Moreover, in the followoing we discuss the explicit foumulae of Fourierexpnasion or spherical functions, mainly we conider only such f that has a scalarK-type, i.e., there is a continuous unitary character

χm : k =(

A B−B A

)∈ K 7→ det(A +

√−1B)m ∈ C∗

such that f(xk) = χm(k)f(x).

2.2 L2-space and spectral decomposition

The other standard way to control the automorphic forms at infinity is to con-sider the L2-space on Γ\G. Let us recall a fundamental result on the spectraldecomposition of the L2-space L2(Γ\G), which is a unitary G-module by theright quai-regular action: ϕ(x) 7→ ϕ(xg) (x, g ∈ G). Since an algebraic real Liegroup G is of type I in the sense of C∗-algebra, the space L(Γ\G) is a direct in-tegral of irreducible unitary representations of G uniquely. The discrete part of

11

this decompostion L2dis(Γ\G) is defiend as the closure of the sum of irreducible

G-subspaces. The continuous part L2con(Γ\G) is the orthogonal complement of

this dicrete part. Hence we have the direct sum decompostion:

L2(Γ\G) = L2dis(Γ\G)⊕ L2

con(Γ\G).

The generalized Plancherel measure on G to describe L2con(Γ\G) is given as

a sum of contribution from various Eisenstein series. An the cuspidal sub-sapce L2

cusp(Γ\G) is a closed subsapce of L2dis(Γ\G). The fundamental theo-

rem of (Selberg-)Langlands in [26] is to describe the orthogonal complement ofL2

cusp(Γ\G) in L2dis(Γ\G) as the residual part comming from the residues of the

poles of Eisenstein series.

2.3 Various cohomology groups

We introduce some variations of cohomology of Γ which are mutually correlated:(i) The cuspidal cohomology group:

Hicusp(Γ, E) := Hi(g,K; E ⊗C L2

cusp(Γ\G)),

where L2cusp(Γ\G) is the closed subspace of L2(Γ\G) consisting of cusp forms,

which decomposes discretely into irreducible unitary G-modules with finite mul-tiplicities.(ii) The square-integrable cohomology group:

Hi(2)(Γ, E) := Hi(A∗(2)(Γ\X, E)),

where (A∗(Γ\X, E), d) is the complex consisting square-integrable forms. Asshown by A. Borel, this space is isomorphic to

Hi(g, K; E ⊗C L2dis(Γ\G)),

where L2dis(Γ\G) is the discrete part of L2(Γ\G), that is the closure of the sum

of the closed irreducible subspaces of L2(Γ\G).(iii) The interior cohomology group:

Hi!(Γ, E) := ImHi

c(Γ\X, E) → Hi(Γ\X, E),

where Hic(∗, ∗) means the cohomology group with compact supports.

Fortunately for G = Sp(2,R) and i = 3, all these cohomology groups areisomorphic and hava a similar decomposition as the cocompact case, replacingL2(Γ\G) by L2

dis(Γ\G) or by a smaller space L2cusp(Γ\G). This was shown by

Oda-Schwermer ([42], Formula (4) in p. 488) for the constant coefficient case,and for the twisted case coefficient cases it is handled esaier. Since the K-finitepart of L2

cusp(Γ\G) is identical with the space of cusp forms Acusp(Γ\G), wehave

H3! (Γ, El) = H3

cusp(Γ, El) = H3(2)(Γ, El).

Moreover these space have a similar decompostion as the case of cocompactsubgroups, replacing C∞(Γ\G) by the space of cusp forms Acusp(Γ\G) to formthe intertwining spaces

Hom(g,K)(π±±l+ρ,Acusp(Γ\G)⊗ El).

12

Similarly as the cocompact case, we introduce 4 spaces:

S++l (Γ) := f ∈ Acusp(Γ\G)⊗W(−l2−3,−l1−3) |

(i) f(xk) = τ(−l2−3,−l1−3)(k)f(x),∀x ∈ G, ∀k ∈ K(ii) D++f = 0

S+−w1(l+ρ)−ρ(Γ) := f ∈ Acusp(Γ\G)⊗W(l2+1,−l1−3) |

(i) f(xk) = τ(l2+1,−l1−3)(k)f(x), ∀x ∈ G,∀k ∈ K(ii) D+−f = 0

S−+w2(l+ρ)−ρ(Γ) := f ∈ Acusp(Γ\G)⊗W(l1+3,−l2−1) |

(i) f(xk) = τ(l1+3,−l2−1)(k)f(x), ∀x ∈ G,∀k ∈ K(ii) D−+f = 0

S−−w3(l+ρ)−ρ(Γ) := f ∈ Acusp(Γ\G)⊗W(l1+3,l2+3) |(i) f(xk) = τ(l1+3,l2+3)(k)f(x),∀x ∈ G, ∀k ∈ K(ii) D−−f = 0

Here D±,± are the Dirac-Schimd operators.As we see in the subsection 3.4, the definition of the Dirac-Schmid operator

is independent of Γ. Therefore it is the same as the case of cocompact Γ.Now we have the analogue of Eichler-Shimura isomorphism:

Theorem 2.1

H3! (Γ, El) = ⊕±,±Hom(g,K)(π±±l+ρ,Acusp(Γ\G)⊗ El)

= S++l (Γ)⊕ S+−

w1(l+ρ)−ρ(Γ)⊕ S−+w2(l+ρ)−ρ(Γ)⊕ S−−w3(l+ρ)−ρ(Γ).

Here the proof of the second equality of the above theorem is the immediateconsequence of the characterization of the discrete series representations by theirminimal K-types (cf. the subsection 3.3).

One important achievement in 80’s on the L2-cohomology is the proof ofZucker’s conjecture. We may refer the artitlce Zucker [53] for the survey of thishistory. We should mention the project of (Harder-)Schwermer to describe the”difference” between Hi

cusp(Γ\G,E) and Hic(G,A(Γ\G; E)) by using Eisenstein

series ([47]).

13

3 The representations of the discrete series ofSp(2,R)

A representation of discrete series of a real semisimple Lie group G is a bothtempered and cohomological representation. The investigation of this very im-portant class of unitary representations are initiated by Harish-Chandra in later’50’s. The fundamental results are all obtained by him: the equivalence withsquare-itegrability, the criterion of existence, the character formula on com-pact Cartan subgroups, etc. Realization (Okamoto-Narashiman, Schmid), thecharacter formula (Hirai, Schmid) K-type theorem (the Blattner conjecture,(Hecht)-Schmid). These deep results are not yet used for geometry and arith-metic of automorphic forms.

In general, let G be a locally compact topological group which satisfies thesecond axiom of countability; let G be the unitary dual of G, i.e., the set of uni-tary equivalence classes of irredicible unitary representaions of G. As desribedin Dixmier [3] by the C∗-algebra of G, G under the hull-kernel topology is alocally compact Baire space.

For a semisimple Lie group G, which is postliminaire, the abstract Plancherrltheorem is vaid (due to Segal): fix a Haar measure dG on G, then there existsunique positive measure µ on G such that

∫

G

|f(x)|2dG(x) =∫

G

tr(π(f)π(f)∗)dµ(π)

for all f ∈ L1(G) ∩ L2(G).The support of the Pancherel measure µ on G is a closed subset Gtemp, and

those unitary representations belonging to this are called tempered representa-tions.

3.1 Definition

For a general semisimple Lie group G with finite center, the unitary represen-tations of G belonging to the discrete series are defined as follows:

Definition(i) Let G be the unitary dual of G with the hull-kernel or the Fell topology.Then π is of discrete series, if and only if π is isolated in G.(ii) The follwing was shown by Harish-Chandra:

Theorem 3.1 π is of discrete sereis, if and only if π is square-integrable, i.e.,

∀v, w ∈ Hπ, the matrix coefficient cv,w(g) = (π(g)v, w)Hπ ∈ L2(G).

(iii) Let us consider L2(G) as a G × G-bimodule via the left and right regularrepresentations. Then we have the Plancherel formula:

L2(G) =∫ ⊕

Gtemp

πλ £ π∗λ dµ(λ).

Here µ is the Plancherel measure which has support only on the temperedspectre Gtemp of G. Then for discrete series πλ, we have µ(πλ) > 0 and thisvalue is proportional to the formal degree of πλ.

The following fundamental result is also due to Harish-Chandra:

14

Theorem 3.2 (Harish-Chandra) The discrete spectre GDS is non-empty, if andonly if rank-K = rank-G, i.e., G has a compact Cartan subgroup T .

3.2 Parametrization of the discrete series

There is also the parametrization of the discrete series by Harish-Chandra, byusing the set of dominant integral weigts on T .

We discuss on the special case G = Sp(2,R). A compact Cartan subgroupT of G in K is given by

T =

cos θ1 sin θ1

cos θ2 sin θ2

− sin θ1 cos θ1

− sin θ cos θ2

| θ1, θ2 ∈ R

.

The set LT of unitary characters of T is parametrized by pairs of integers(l1, l2) ∈ Z2 and the set of dominant weights is given by

L+T := (l1, l2) ∈ L | l1 ≥ l2.

The irreducible finite-dimensional representation of K with highest weight (l1, l2) ∈L+

T is denoted by (τ(l1,l2),W(l1,l2)). Let

Ξ := l = (l1, l2) ∈ L+T | l regular, i.e., l1 6= 0, l2 6= 0, l1 6= ±l2.

We note here that ρ the half-sum of positive roots for (g, t) is given by

ρ = 122e1 + (e1 + e2) + 2e2 + (e1 − e2)

= 12(2, 0) + (1, 1) + (0, 2) + (1,−1) = (2, 1).

Theorem 3.3 (Harish-Chandra) For G = Sp(2,R), there is a bijection πλ ∈GDS ↔ λ ∈ Ξ. Moreover the set Ξ is written as the sum of 4 Weyl chambers:

Ξ = Ξ++ ∪ Ξ+− ∪ Ξ−+ ∪ Ξ−−

withΞ++ := (λ1, λ2) ∈ Ξ | λ1 > λ2 > 0Ξ+− := (λ1, λ2) ∈ Ξ | λ1 > 0 > λ2, l1 + l2 > 0Ξ−+ := (λ1, λ2) ∈ Ξ | λ1 > 0 > λ2, l1 + l2 < 0Ξ−− := (λ1, λ2) ∈ Ξ | 0 > λ1 > λ2.

For a given λ = (l1, l2) ∈ Ξ satifying l1 > l2 > 0, there are 4 parameters:

λ++ = (λ1, λ2) ∈ Ξ++, λ+− = (λ1,−λ2) ∈ Ξ+−,λ−+ = (λ2,−λ1) ∈ Ξ−+, λ−− = (−λ2,−λ1) ∈ Ξ−−

with the same infinitesimal characters χπλ∗∗ : Z(g) → C.

15

3.3 The minimal K-types

From some time, there was the Blattner conjecture to describe the multiplicitiesof K-types a given discrete series. This was finally proved by Hecht-Schmid [7].Some what earlier the minimal K-type τ whose highest weight is nearest to theorigin that occurs with multiplicity one is known to exist. We recall this minimalK-type in the case G = Sp(2,R). For a given Harish-Chandra parameter λ ∈ Ξ,we choose the postive root sytem Φ+(λ) which is positive with respect to λ. Letρc be the half-sum of the compact positive roots which is independent of thechoice of λ, and let ρn(λ) be the half-sum of non-compact postive roots inΦ+

nc(λ). Then we have the following (cf. Schmid [46]).

Proposition 3.1 Set µ = λ−ρc +ρn(λ). Then τµ is the minimal K-type of thediscrete series πλ, and for any β ∈ Φ+

nc(λ), the K-type τµ−β does not occur inπλ. Moreover the condition that such K-type exists, which is far enough from thewalls of the Weyl chambers, characterises the discrete series πλ infinitesimally.Namely, if τµ is far enough from the walls of Weyl chambers, and τµ occurswith multitplicity one in some irreducible (g,K) module π, and if and for anyβ ∈ Φ+

nc(µ), the K-type τµ−β does not occur in π. Then π is isomorphic to the(g,K)-modules of the discrete series πµ+ρc−ρn(µ).

Here is the table of the minimal K-types µ of the discrete series for Sp(2,R).Here λ = (λ1, λ2) are Harish-Chandra parameters.

type of parameter λ λ ∈ Ξ++ λ ∈ Ξ+− λ ∈ Ξ−+ λ ∈ Ξ−−

the minimal K-type µ λ + (1, 2) λ + (1, 0) λ− (0, 1) λ− (2, 1)

Remark 1 The condition ”if τµ is far enought from the walls of Wely chambers”is not necessary for G = Sp(2,R) case. We know the infinitesimal character ofthe discrete series representation in question, and also the composition series ofthe principal series representation with this infinitesimal character explicitly.

3.4 Realization of discrete series reprentations

There are realizations of the discrete series represenations in the spaces ofsquare-integrable ∂-cohomology groups over G/K or over G/T with values inholomorphic vector bundles associated with irreducible finite-dimensional rep-resentaions of K or with unitary chatacters of T (cf. [37], [45]), which areemployed to prove a conjecture of Langlands. And this smooth realizationwas later generalized for smooth realization of cohomological represenations byWong [51].

But for explicit calculation of the A-radial part of the functions belongingto the minimal K-type, the following realization of the discrete series represen-tations, utilizing the Dirac-Schmid operator is quite essential.

For a finite dimensional continuous representation (τ, W ) of K, we set

C∞τ (G;K) := C∞(G)⊗WK .

Here K-invariant part ∗K is considered with respect ot the right quasiregularaction of K on C∞(G) and τ on W . Let (Ad, pC) and (Ad, p∗C) be the adjointrepresentation of K and its contragradient representation, which are isomorphic

16

by the non-degenerateness of the restriction of the Killing form to p. Then wehave an isomorphism of K-modules

(Ad, pC) ∼= (Ad, p∗C) ∼= ⊕β∈Φnc(l)(τβ ,Wβ)

with Φnc the set of non-compact roots in Φ(g, t). Choose a basis xi in p andits dual basis ξi in p∗, then we can define the gradient operator

∇ : C∞τ (G; K) → C∞τ⊗(Ad∗, p∗)(G;K)

by∇f :=

∑

i

Rxif ⊗ ξi

with Rxi the right action of p ⊂ g on C∞(G).Now choose a dominant integral weight l of T to have the associated irre-

ducible representation (τl,Wl) of K. We assume that l satisfies the regular-ity condition such that there is unique positive system of non-compact rootsΦ+

nc(l), which is positive with respect to l (and compatible with the given sys-tem of compact positve roots Φ+

c ). Then we have the associated decompostionof p∗C = p∗+(l)⊕ p∗−(l) with

p∗+(l) := ⊕β∈Φ+nc(l)

τβ , and p∗−(l) := ⊕β∈Φ+nc(l)

τ−β .

Then from the tensor product

(τl ⊗Ad∗, Wl ⊗ p∗C) ∼= ⊕β∈Φnc(τl+β , Wl+β),

there is a projector of K-modules

pr− : Wl ⊗ p∗ → Wl ⊗ p∗−(l) ∼= ⊕β∈Φ+nc(λ)(τl−β , Wl+β).

Lastly we consider the compostion of the gradient operator ∇ with the projector

C∞τl⊗Ad∗(G; K) → C∞τl⊗Ad∗−(G; K),

to get an operator Dl between the smooth left G-modlues :

C∞τl(G; K) → C∞τl⊗Ad∗−

(G; K).

Here Ad∗− means the K-module (Ad∗−, p∗−(l)). This operator, which we call theDirac-Schmid operator, is an elliptic operator under some condition on l, whichis satisfied when l is the minimal K-type of a discrete series with paramater λpostive with respect to Φ+(l).

Theorem 3.4 Let τl be the contragradient representation of the minimal K-type µ = λ − ρc + ρn of the discrete series with Harish-Chandra parameter λ.Then the kernel space Ker(Dl) of the operator is non-zero and by the left quasi-regular action, it gives the contragradient discrete series representation π∗λ.

Remark 1. It is easy to understand the ”meaning” of the above theorem. Foreach discrete series πλ, it occurs as the outer tensor product π∗λ £ πλ in the(G×G)-birepresentation L2(G) uniquely. Consider the intertwining space

HomG(πλ, L2(G))

17

to absorbe the right G-action and to realize π∗λ as this intertwining space. Andconsider further the evaluation map

HomG(πλ, L2(G))⊗ πλ → L2(G).

The restriction of this evaluation map to the minimal K-type τµ gives canoni-cally

evK : HomG(πλ, L2(G)) → L2(K)⊗ τ∗µK .

Here the superscript ∗K means the K-invariant part. Then the conditionof the annihilation p−(λ) · τµ = 0 implies that the image of the inducedevaluation map evK is contained in the kernel of the Dirac-Schmid operator.The surjectivity follows from the latter part of Proposition 3.1. (cf. Hotta-Parthasarathy [12]).Remark 2 Original proof was done under stronger regularity condition, to assurethe characterization of the minimal K-type of the discrete series to assure theellipticity of Dl. This is unneccesary here for G = Sp(2,R).

3.5 The Dirac-Schmid operators on automorphic forms

Let Acusp(Γ\G) be the space of automorphic cusp forms on G with respect toΓ. Then for a given discrete series representation πλ, we consider the evaluationmap

evΓ : Hom(g,K)(πλ,Acusp(Γ\G))⊗ πλ → Acusp(Γ\G).

The image of this is the πλ-isotypic component inAcusp(Γ\G). Take the minimalK-type i : Wµ 7→ πλ to have the indued K-homomorphism by restriction:

evΓ : Hom(g,K)(πλ,Acusp(Γ\G))⊗Wµ → Acusp(Γ\G),

or passing to the adjoint as K-modules, we have

evΓ,K : Hom(g,K)(πλ,Acusp(Γ\G)) → Acusp(Γ\G)⊗W ∗µK .

Discussing similarly as the previous remark 1, we can show that the image ofthis belongs to the kernel of the Dirac-Schmid operator

pr− · ∇ : Acusp(Γ\G)⊗W ∗µK → Acusp(Γ\G)⊗W ∗

µ ⊗ p−(λ)K ,

which we also denote by D±,±. The surjectivity is obtained similarly as the lastremark 1.

18

4 Some general aspect on Fourier expansion ofautomorphic forms

At first sight it seems to be a rather trivial problem to have Fourier expansionsof automorphic forms. But after a serious study, one find that it is a difficultproblem to find some reasonable form of Fourier expansion for general automor-phic forms in the sense of Harish-Chandra [5], with respect rational parabolicsubgroups. Even for the rather small grouo SU(2, 1) it was worked out ratherrecently by Ishikawa [19]. And for holomorphic automorphic forms on a symmet-ric domain G/K, though there were Fourier-Jacobi expansions with respect tovarious rational maximal parabolic subgroups P , the Fourier expansion with re-spect to general parabolic subgroups including the minimal parabolic subgrouphas settled quite recently in a satisfactory manner by Narita [38] for G/K be-ing tube domains. The well-known theory of Fourier expansion on GL(n) byH. Jacquet, I. Piatetski-Shapiro, and J. Shalika is basically valid only for themaximal parabolic subgroup of type (1, n− 1) utlizing the semi-global informa-tion of Whittaker functions, and moreover Whittaker functions are defined withrespect to the minimal parabolic subgroup of GL(n) and linear characters of itsunipotent radical.

4.1 General formulation of the problem

Before discussing the Fourier expansion of automorphic forms with respect spe-cific parabolic subgroups, we consider the general setting of Fourier expansionwith respect to a ”good” closed subgroup R of G defined ove Q. In the beginningwe assume that

(H1) : (R ∩ Γ)\R is compact,

to avoid the trouble of the continuous spectrum. For r ∈ R and g ∈ G, weconsider a function f(rg) of two variables (r, g) ∈ R × G for a vector f in theimage of the evaluation map

evπ : Hom(g,K)(Mπ,A(Γ\G))⊗Mπ → A(Γ\G)

for some fixed (g,K) module Mπ.Fix the second variable for a while to regard f(rg) as a function in r. Then

it is real analytic in r which is left (R ∩ Γ)-invariant and right R ∩ (g−1Kg)-invariant. Since it is not of Z(r)-finite in general, it is an infinite sum

f(rg) =∑

η∈R

f(rg)[η]

of automorphic forms on R via the spectral decomposition

L2((R ∩ Γ)\R) = ⊕η∈RI(η)

with the η-isotypic component

I(η) ∼= HomR(Vη, L2((R ∩ Γ)\R))⊗ Vη.

Here Vη is the representation space of η.

19

For each fixed η, let Bη = βη,i1≤i≤m(η) be a basis of the finite-dimensionalvector space HomR(η, L2((R ∩ Γ)\R)) of dimension m(η). Then

f(rg) =m(η)∑

i=1

evη(βη,i ⊗ vη,i)(r)

with some vectors vη,i = vη.i(f ; g) ∈ Vη depending on f and g. Here evη is theevaluation isomorphism:

evη : HomR(η, L2((R ∩ Γ)\R))⊗ Vη∼= L2((R ∩ Γ)\R)[η].

Computing the function f(rr′g) (r, r′ ∈ R, g ∈ G) of 3 variables in two ways,we have an intertwining property:

vη,i(f ; rg) = η(r)vη,i(f ; g) (r ∈ R, g ∈ G).

Therefore for fixed f , the function

g ∈ G 7→ vη,i(f ; g) ∈ Vη

defines an element of the representation space of IndGR(η). Moreover the associ-

ationf ∈ Mπ ⊂ A(Γ\G) 7→ vη.i(f ; g) ∈ IndG

R(η)

is a G-homomorphism. Thus each vη,i (i = 1, · · · ,m(η)) is considered as anelement of the intertwining space Hom(g,K)(Mπ, IndG

R(η)).If we have ”the multiplicity-free theorem” Hom(g,K)(Mπ, IndG

R(η)) ≤ 1, thenthere is the unique normalized element vη in this space, and each vη,i is writtenin the form

vη,i(f ; g) = cη.i(f)vη(f ; g)

with a constant (i.e., the Fourier coefficient) cη,i(f) which is independent of gand i.

When R is the unipotent radical N of a minimal parabolic subgroup P ofG, and η a unitary character of N , then all the above produre is valid and vη

is called the Whittaker functional. For a infinite-dimensional η of M , or moregenerally for general parabolic subgroups P , the multiplicity-free property isnot valid even for the characters η of the unipotent radical R = N of P . Inthis case, we have to take as R a subgroup between P and N , which is in manycases (a metaplectic cover of) the semidirect product of N and the stablizer Sof a represenation ξ of N in the Levi-part L of P .

Anyway we are naturally lead to the follwoing problem of spherical functions.

4.2 The local problem: spherical functions

Let us formulate our problem on generalized spherical functions. Let R be aclosed subgroup, and η a unitary irreducible representation of R. Form theC∞-induction C∞-InG

R(η). Given a smooth represenattion π of G, or a (g,K)-module (π, Hπ). Then we can consider the intertwining space

I(R,η)(π) := Hom(g,K)(π, C∞-IndGR(η)).

We hope the situation when R is ”large enough” and η is ”small enough”, tosatisfy the following conditions:

20

(i) For the split Cartan subgroup A, we have a double coset decompostionG = RAK.

(ii) The dimension of I(R,η)(π) is finite.

In the context of ordinary Fourier expansion, it is enough to consider whenR is a subgroup of a parabolic subgroup P containg the unipotent radical N ofR. But it is also important to consider the case when R is a reductive subgroupof G, as we see in the later section 6.

4.3 The PJ principal series representations of Sp(2,R)

We should review the results on the spherical functions belonging to the dis-crete series represenations of Sp(2,R). But for holomorphic discrete series, thisproblem is rather easy, and more or less it is known classically. For the largediscete series one need to prepare some more terminiolgy to specifty the basis ofirreducible represenations τl of K (especially for the minimal K-type τ). Thisrequires much more space.

In place of the large discrete series themselves, we will review the results onthe spherical functions belonging to the PJ principal series represenation, whichis closely related to the large discrete series represenations of Sp(2,R) in twoways: (i) the large discrete series are embedded in a PJ principal series; (ii)the K-types of the large discrete series are translations of the K-types of thePJ principal series. As a result, the forms of spherical functions of the largediscrete series are quite similar to those of the PJ principal series.

4.3.1 The definition of PJ principal series

Here is the definition of the PJ principal series. Firstly, we consider th standardparabolic subgroup PJ associated with the long root 2e2 = (0, 2) in the set ofpostive roots

∆ = e1 − e2 = (1,−1), 2e2 = (0, 2).Its unipotent radical NJ is the Heisenberg group of dimension 3 with Lie algebra

nJ = ge1+e2 ⊕ g2e1 ⊕ ge1−e2 .

Thus the half sum of postive roots ρJ is given by (2, 0) = 12 (e1+e2+2e1+e1−e2).

The split componet AJ is associated with the Lie algebra aJ = RH1 with

H1 = diag(1, 0,−1, 0).

The group MJ is the product of the centralizer of AJ in K and the connectedcomponent M0

J which correponds to the Lie algebra

mJ = g2e2 ⊕ g−2e2 ⊕RH2.

Since the group MJ is isomorphic to the product ±1×SL(2,R), the discreteseries represenation of MJ is the tensor product ε⊗D±

k of a character ε of ±1and a discrete series represenation D+

k or D−k of SL(2,R) with minimal K-type

χk : r(θ) =(

cos θ sin θ− sin θ cos θ

)∈ SO(2) 7→ exp(kθ) ∈ U(1)

or χ−k : r(θ) 7→ exp(−kθ) ∈ U(1).

21

Fix a complex-valued linear form νJ ∈ HomR(aJ ,C) which is identified withthe complex number νJ(H1), and the associated quasi-character

eνJ+ρJ : a ∈ AJ 7→ exp(ν + ρJ)(log a) ∈ C∗,

and form the outer tensor product (ε ⊗ D±k ) £ eν+ρ to get a represenation of

the product MJAJ . Via the canonical isomorphism PJ/NJ∼= MJAJ , we can

regard this as a representation of PJ . On the Hilbert space

IndGP ((ε⊗D±

k ) £ eν+ρ) :=f : G → HD±

k

∣∣∣f(nmax) = (ε⊗Dpm

k )(m)eν+ρ(a)f(x),for a.e.n ∈ NJ ,m ∈ MJ , a ∈ AJ , x ∈ G∫

K‖f(y)‖H

D±k

dy < ∞

The action of G is by the right translation. We denote the obtained Hilbertrepresentation by πPJ ;ε⊗D±

k ,νJ. Since the cases D+

k and D−k are mutually ∗-

conjugate, we consider only the case of D+k .

When ε(−1) ·(−1)k = 1, we call this represenatation even PJ principal seriesrepresentaion, and otherwise odd.

4.3.2 The K-types of the PJ principal series

We consider an even PJ principal series representation π± = πPJ ;ε⊗D±k ;νJ. In

order to describe the K-types of even π±’s, we introduce some notation first.Notation For a dominant integral weight (l1, l2) ∈ L+

T with parity condition(−1)l1+l2 = +1, we put

(l1, l2) =

(l1, l2), if (−1)li = ε (i = 1, 2);(l1 − 1, l2 + 1), if (−1)li = −ε. (i = 1, 2)

.

Proposition 4.1 If (−1)l1+l2 = +1, the multiplicity [π± : τ(l1,l2)] of τ(l1,l2) ∈ Kin π+ (resp. π−) is given by

[π+ : τ(l1,l2)] =12(supk, l1 − supk, l2) + 1

and[π− : τ(l1,l2)] =

12(inf−k, l1 − inf−k, l2) + 1.

If (−1)l1+l2 = −1, then [π± : τ(l1,l2)] = 0.

4.3.3 The annihilator of the corner K-type of even PJ principal series in U(g)

We have[π+ : τ(l1,l2)] = 1 and [π− : τ(l1,l2)] = 1,

respectively. Each K-type τ(±k,±k) in π± is called the corner K-type, respec-tively. Here imagine the ”picture” of the constituents of the K-types of eachπ±.

Let v+ or v− be the non-zero vector, unique up to scalar multiple, which isin the image of the non-zero K-homomorphism τ(k,k) → π+ or τ(−k,−k) → π−,

22

respectively. Then we have some element of degree 2 in the universal envelopingalgebra S(p−) = U(p−) (resp. S(p+) = U(p−)), which annihilates the image ofthis homomorphism. The holomorphic part p+ (or the anti-holomorphic partp−) of pC with respect to the complex structure Ad(ι) given by an element ι oforder 8 in the center of K, has a basis X+11, X12, X22 (or X−11, X−12, X−22,resp.) specified as follows, by using the matrix units Eij ∈ M4(C):

X±11 := E11 − E33 ±√−1(E13 + E31),

X±22 := E22 − E44 ±√−1(E24 + E42),

X±12 := 12 (E12 + E21 − E34 − E43)± 1

2

√−1(E14 + E23 + E32 + E41).

Proposition 4.2 We have

(X−11X−22 −X2−12)v(k,k) = 0, or (X+11X+22 −X2

+12)v(−k,−k) = 0,

respectively. Moroever since π± are quasi-simple, they have the infinitesimalcharacters χπ± : Z(g) → C, and in particular we have equations:

C − χπ±(C)v(±k,±k) = 0

for the Casimir operator C.

Remark 1 The second statement is a standard fact. The first statement in theabove proposition is not formulated in [31] so explicitly, but this is the core rea-son to have one of the main differential equations in [31]. The interesting point isthe radial part of this operator gives the Euler-Darboux operator with respect tonatural coordinates in various realizations or models of our PJ principal series.Remark 2 Logically speaking we do not need it, but it would be helpful tounderstand PJ prinicipal series to know the fact:

Proposition 4.3 The large discrete series is embeddable in a PJ principal se-ries.

The details and application of the above proposition will be discussed in aforthcomming paper.

23

5 The known facts on Fourier expansions of au-tomorpphic forms on Sp(2,Z)\Sp(2,R)

We discuss some of the known results on the Fourier expansion on Γ\Sp(2,R)up to now, where Γ is Sp(2,Z) or one of its congruence subgroups.

5.1 The Fourier expansion with respect to P0

The Fourier expansion with respect to the minimal parabolic subgroup P0 ismost difficult to handle. We do not have satisfactory results for general au-tomorphic forms f and for general infinite-dimensional represenations η of theunipotent radical N0. Usually only the case of linear characters η are discussed.The case of holomorphic discrete seriesIn this case, there are no terms terms corresponding to the characters η ofN . This fact is expressed as the vanishing of the Whittaker models in therepresentation-theoretic formulation of the problem. Heuristically speaking, thesmallness of the holomorphic discrete series representations (i.e., their Gelfand-Kirillov dimension is 3) strikes out the ”enough room” for the existence of theassociated Whittaker functions.

In this case we have to consider the terms for infinite-dimensional η ∈N0. We have some partial result. As we have alreay seen in the subsec-tion2.3, the (+,+)-compoent of H3

! (Γ\G/K, El) is realized by the space ofholomorphic automorphic cusp forms belonging to the weight (l1 + 3, l2 + 3),i.e., vector-valued automorphic cusp forms belonging to the automorphy factor

detl2+3⊗syml1−l2(CZ + D) for γ =(

A BC D

)∈ Γ. For holomorphic modu-

lar forms, the Fourier expansion along the minimal parabolic subgroup P0 wasdiscussed by Narita [38]. Similarly the (−,−)-component corresponds to theanti-holomorphic automorphic forms.

The case of the large discrete series representationsFor the (+,−)- or (−, +)-component, we have to do something new. The ele-ments in this component generate large discrete series representations. The firstproblem is to find the Whittaker functions belonging to the large discrete series.This was done by Oda [41] for the large discrete series and by Miyazaki-Oda[31] for PJ -principal series.

We see the explicit formula of the A-radial part hW of the spherical functionϕ = I(v(k,k)) with the corner K-type in the Whittaker model Im(I) of the PJ

principal series π± for a Whittaker functional I ∈ Hom(g,K)(π±, IndGN0

(η)).

5.1.1 The holonomic system of A-radial part

Proposition 5.1 ([31]) Let hW (a1, a2) := h(diag(a1, a2, a−11 , a−1

2 ) be the A-radial part of the Whittaker function ϕ(g) with the corner K-type, belonging tothe PJ -principal series πPJ :D−

k ,νJ. Then this satisfies a system of equations:

[∂1∂2+(4πc3a22−k)∂1−(k+1)∂2−4π(k+1)c3a

22+k(k+1)+4π2c2

0(a1

a2)2]hW (a) = 0

[∂21 + ∂2

2 − 4∂1 − 2∂2 − 8π2c20(

a1

a2)2 − 16π2c2

3a42 + 8πkc3a

22]hW (a) = 0.

24

Here ∂i = ai∂∂ai

(i = 1, 2).

5.1.2 Integral expression

Theorem 5.1 (i) Let ϕ := I(v(−k,−k))|A be a Whittaker function if πPJ ;D−kwith K-type (−k,−k). Then it has an integral expression

ϕ(a1, a2) = ak+11 ak

2 exp(−√−1η2e2a22)

× ∫∞0

t−k+ 12 W0,νJ

(t) exp(− t2

32√−1η2e2a2

2+

8√−1η2

e1−e2η2e2a2

1

t2 )dtt .

(ii) (Moriyama) Up to constant multiple

hW (a1, a2) = (2π√−1)−2

∫Re(s)=σ1

∫Re(s)=σ2

MνJ ,ka−s11 a−s2

2 ds1ds2

(σ1 > σ2, Re(σ1 + σ2 ± νJ − k + 1) > 0)

with

MνJ ,k(s1, s2) = Γ(s1 + s2 + νJ − k + 1

2)Γ(

s1 + s2 − νJ − k + 12

)Γ(s1

2)Γ(−s2

2)

Remark. And this result is utilized by Moriyama [35] to show analytic contin-uation and the functional equation for spinor L-functions of automorphic formsbelonging to the large (or generic at R if you prefer this terminology) disreteseries represenations.

The Whittaker functions are defined only for (linear) characters of N0. Thegroup N0 has infinite-dimesional represenattions in general. We know littleabout the generalized Whittaker functions with respect to η of dim η = ∞.

Open Problem Investigate the generalized Whittaker models or Whittakerfunctions for infinite-dimensional η ∈ N0.

5.2 The Fourier expansion with respect to the Siegel parabolicsubgroup PS

5.2.1 The naive formulation of the problem

Let

PS :=(

A B0 D

)∈ G|A,B, D ∈ M2(Q)

be the rational standard maximal subgroup associated with the short root, whichis called Siegel parabolic subgroup.

An element of this group is written as(

A B0 D

)=

(A 00 tA−1

)·(

12 B′

0 12

)

(A,D ∈ GL(2,R), tAD = 12, B′ = tB′ ∈ M2(R))

and any element of the unipotent radical NS is given as(

12 B0 12

)(B = tB ∈ M2(R)).

25

In this case NS is abelian, hence all the irreducible unitary representations ofNS is abelian, i.e., characters. Therefore the function f(nSx) in nS ∈ NS hasthe associated Fourier expansion of the form:

f(nSx) =∑

η∈ N/(Γ∩N)

η(nS)SW fη (x)

The unitary characters of NS trivial on NS,Γ = Γ ∩NS can be expressed as

ηT (nS) = exp2πitr(T ·B(nS)) for nS =(

1 B(nS)0 1

)

for some symmetric matrix T = tT ∈ M2(Q) with some integrality conditionfor NS,Γ. Therefore the naive form of the Fourier expansion of an automorphicform f on Γ\G along PS is given by

f(nSg) =∑

T=tT∈ (NS/NS,Γ)

exp2πitr(T ·B(nS))SWfT (g),

where

SWfT (g) =

∫

NS,R/NS,Γ

exp−2πitr(T ·B(nS))f(nSg)d(nS),

with d(nS) is the normaized Haar measure on NS/NS,Γ. Since G = PS ·K, if weassign the special K-type to f from the right side, to know f it succies to knowthe restriction f |PS . Meanwhile we have the Levi decompostion PS = LS ·NS .Hence to know

SWfT (pS) = SWf

T (nS · lS) = exp2πitr(T ·B(nS)b)SWfT (lS),

it suffice to know its restriction to LS . But the homogeneous space LS/(LS∩K)still has real dimension 3, and the differential equations comming from thecenter Z(g) of U(g) are not enough many to give a holonomic system on thishomogenous space in general.

Note that the case of holomorphic automorphic forms f which are classicalSiegel modular forms is exceptional, because in this case the Cauchy-Riemanncondition of many varaibles make the represenation spaces of π(+,+), π(−,−) quitesmall (i.e, the Gelfand-Kirillov dimension of the automorphic representationsgenerated by f is 3).

In general, to have tractable form of Fourier expansion, we need furtherexpansion of SWf

T (lS).

Assume that T 6∼(

0 11 0

), and put

St(ηT )0 = SO(T ) = lS ∈ GL(2,R) = LS(R) | tlST lS = T0.Then we have LS =

⋃b∈SO(T ) bAb−1 for

A = diag(a1, a2, a−11 , a−1

2 ) | ai ∈ R>0 (i = 1, 2).If T is definite, then SO(T ) is isomorphic to SO(2), a compact abelian group,whose characters are given by

χm : rθ ∈ SO(2) 7→ exp(imθ) ∈ U(1).

26

Here rθ ∈ SO(2) is the rotation of the angle θ on the Euclid plane R2. Thenwe can expand SWf

T (lS) further as

SWfT (σlSσ−1) =

∑

m∈Z

χm(σ)SWfT,m(lS) (σ ∈ SO(T )).

Then finally we have to ask whether we have enough data to specify SWfT,m|A

for each pair T, m.

5.2.2 The Siegel-Whittaker functions

Apart from the naive formulation of the problem, we discuss the actual for-mulation of the problem to specify the functions SWf

T,m|A in the context ofrepresentation theory.

For T = tT ∈ M2(R), we can associate the subgroup

SO(T ) = St(ηT )0

similarly as in the previous subsection. Assume that T is definite from now onin this subsection. Then the group SO(T ) is compact and isomorphic to SO(2).We can define the unitary characters χm for each m ∈ Z as above. Then forthe semidirect product

Rη = NS · SO(T ),

we can associate the represenation η £ χm, the twisted outer tensor product.Then the local problem at the real place is to investigate the intertwining

spaceHom(g,K)(π, IndG

Rη(η £ χm))

for the standard representations or all the irreducible representations π of thepair (g,K). The following is known until now.

Theorem 5.2 (Niwa, Miyazaki, Ishii) Let π be a represenation of the discreteseries, or principal series, or PJ -principal serie (i.e., only the generalized prin-cipal series obtained by parabolic induction with respect to PS is not handled).Then, the dimension of the intertwining space

Hom(g,K)(π, IndGRη

(η £ χm))

is at most the half of the Bernstein degree of the representation in question (i.e.,2, 4, 2, respectively in this case).

Moreover if we replace the target space IndGRη

(η £ χm) of the intertwiningoperator by the subsapce IndG

Rη(η£χm)rapid consisting of vectors which decrease

rapidly on A at infinity, then the dimension of the intertwinig operator :

Hom(g,K)(π, IndGRη

(η £ χm)rapid)

is at most one (the multiplicity-free theorem); Furthermore, we have explicitintegral expression of the A-radial part of the image of some vector in π withsome specified K-types by this unique intertwining operator.

27

5.2.3 Explicit formulae

In this subsubsection, we show somme explicit integral formulae for our ’Siegel-Whittaker functions. Let I be a non-zero intertwining operator in the spaceHom(g,K)(π, IndG

Rη(η £ χm) and let φSW be the A-radial part of the image

I(v(k,k)) of the non-zero vector of τ(k,k)-type in the PJ principal series π+. Herethe elements of A are denoted by daig(a1, a2, a

−11 , a−1

2 ) ∈ A. First we want tohave the holonomic system for φSW (a1, a2).The holonomic system of the A-radial part

Proposition 5.2 ([29]) Let T = diag(h, h) (h > 0). Set

φSW (a1, a2) = (a1a2)k+1 exp(−2πh(a21 + a2

2))hSW (a1, a2).

Then hSW satisfies the holonomic system of rank 4:

[∂1∂2 − a22

a21−a2

2∂1 + a2

1a21−a2

2∂2 + m2 a2

1a22

(a21−a2

2)2 ]hSW (a) = 0

[(∂1 + ∂2 + k − ν)(∂1 + ∂2 + k − ν)− 8πha2

1(∂1 + 1)− 8πha22(∂2 + 1)

]hSW (a) = 0.

with ∂i = ai∂i

∂ai(i = 1, 2).

Here note that the first equation comes from the annihilator X−11X−22 −X2−12 and the second from the eigen-equation with respect to the Casimir op-

erator.

Integral expressionsWe have to solve the above holonomic system. There are three different

integral expressions. The first two are Eulerian, the last is of Barnes’ type. Weshow only the first two here.

Theorem 5.3 (Theorem (7.5) of Miyazaki [29]) Here T = diag(h, h) with h >0. Then the A-radial part of the Siegel-Whittaker function is of the form

b′(a1, a2) = const.e−mπ√−1/2 ak+1

1 ak+22

Γ(2m+1)

∫∞0

tmJm(2π√−1(h(a2

1 − a22))×

2F1( 2−k+2m+ν12 , 2−k+2m−ν1

2 , 2m + 1;−t)e−2πh(a21+a2

2)(t+1)dt

Here Jm is the Besseel function, 2F1 the Gaussian hypergeometric function. Andk, ν1 are the parameter of the representation of the even PJ -principal series.

Here is another integral expression of the same fucntion b′(a1, a2). Basicallythis is obtained in the same way as Gon [4].

Theorem 5.4 Here T = diag(h1, h2) with h1 = h2 = h > 0. Then the A-radialpart of the Siegel-Whittaker function is of the form

b′(a1, a2) = conts.(a1a2)k+1e−2πh(a21+a2

2)c(a1, a2)where

c(a1, a2) = |a21 − a2

2||m|/2 ∫ 1

0t|m|/2−1(1− t)|m|/2−1G(ta2

1 + (1− t)a22)dt

withG(t) = t−

k+|m|+12 et ·W 1

2 (k−|m|−1),±nu12

(2t).

Here W∗,∗ is the W -Whittaker function under the notation of Whittaker-Watson,which decreases very fast at infinity.

28

Open Problem When T is indefinite, St(ηT )0 = SO(1, 1) ∼= R>0 is non-compact. For this we are ignorant about the multiplicity-free property of Siegel-Whittaker models (or fuctions).

5.2.4 Application to automorphic L-functions

The explicit integral formula for Siegel-Whittaker functions has application tohave analytic continuation and the functional equation for the spinor L-functionof automorphic forms, by the method of Andrianov. The original papers ofAndrianov treated the cases of holomorphic Siegel modular forms. The resultwas later extended to the wave forms on the Siegel upper half space of degree2 by Hori [11] utilizing the explicit integral formula of the class one Whittakerfunctions by Niwa [39]. Miyazaki further extended this method to modularforms belonging to the large discrete series [29].

5.3 Fourier expansion along PJ

In this case, we consider a spherical subgroup NJ ⊂ R ⊂ PJ which is now calledJacobi subgroup. We do not yet have global expansion except for the case ofholomorphic Siegel modular forms. The holomorphic case, we have the so-calledFourier-Jacobi expansion.

For the local problem of spherical functions, Hirano ([9], [10]) obtained theexplicit formulae of spherical functions and the multiplicity-free theorem for thecases of the PJ principal series and the discrete series representations.

Open Problem 1 Find a reasonable Fourier expansion utlizing the abovemensioned result of Hirano. Among others, consider how to settle the problemof ”continuous” spectrum adequately.Open Problem 2 There is a yet-another way to have the spinor L-functionfrom automorphic forms on Γ\Sp(2,R) by Kohnen-Skoruppa [24]. ObviouslyHirano’s result should have application if one try to extend the method of [24]for non-holomorphic automorphic forms.

6 Spherical functions with respect to reductivesubgroups

There is one produtive heuristic idea in the theory of automorphic forms, thatis the analogy between the objects involved in parabolic subgroups and those in”hyperbolic” reductive subgroups. The basic credo here is what was done forparabolic subgroups should be also be done analogously for reductive subgroups.

For example as there are Eisenstein series associated with parabolic sub-groups, so there are Poincare series associated with reductive subgroups. Onesuccessful construction by this idea is our result on Green function ([43]). An-other ”example”, which should be cultivated, is the hyperbolic Fourier expansionwith respect to reductive spherical subgroups. The classical work by C. L. Siegelon the Fourier expansion along the tori G2

m → GL(2) is a special case. We donot yet know even the simplest extension of this result to Gn

m → GL(n).

29

6.1 Results on spherical functions with respect to the re-ductive spherical subgroups

There are already some papers on the spherical functions with respect to reduc-tive spherical subgroups of a semisimple algebraic groups.

In their theory of construction of automorphic L-functions for automorphicforms on algebraic groups of type A and type BD, Murase-Sugano, [36] consid-ered such problem. For real groups of type A, Tsuzuki [49] has worked out rathercompletely for the case of the pair (G, R) = (U(n, 1), U(1) × U(n − 1, 1)): inthis important paper, he obtained the multiplicity-free theorem and explicit for-mulae of the radial part of the spherical function ([49], I), the local zeta integral([49], II), and the c-function of the Poisson transformation to have the criterionfor non-vanishing of spherical functions ([49], III). For our group G = Sp(2,R)and R = SL(2,R) × SL(2,R) or R = SL(2,C), Moriyama [32, 33] obtainedexplicit formulae of spherical functions by using Meijer’s G-functions. Hayatahas some results for (G,R) = (SU(2, 2), Sp(2,R) (cf. [6]).

6.2 Construction of Green functions for modular divisorson arithmetic quotients of bounded symmetric do-mains

This subsection is a short review of the joint work with Masao Tsuzuki [43].

6.2.1 Logarithmic Green functions

Given a (smooth, for simplicity) subvariety Y of codimension d in a smoothquasi-projective complex algebraic variety X of dimension n, a current δY , i.e.a differential forms with coefficients in distributions, of type (d, d) on X isdefined by associating the values of the integral

∫

Y

i∗(ω)

for every (n− d, n− d)-type C∞ form ω on X. Here i : Y ⊂ X is the inclusionmap. This is a closed form.

Now, if there is a current g of type (d− 1, d− 1) on X such that

ddcg + δY

is a smooth differential form of type (d, d) on X, then g is called a Green currentfor Y . Here dc = (∂− ∂)/2π

√−1 if we write d = ∂ + ∂ as a sum of holomorphicpart and antiholomorphic part. The form g is not unique.

In the intersection theory of Arakelov type for higher dimensional cases,Gillet-Soule [48] defined Green current of logarithmic type. When the codimen-sion d > 1, this is defined by using the resolution of singularities of Hironaka,but when d = 1, i.e. when Y is a divisor, it is done more directly.

If one defines a Hermitian metric ‖ ‖ on the holomorphic line bundle L =OX(Y ) by putting ‖f‖ = e−g|f | for local section f of L, then the Chern formof L with respect to this metric is given by ddcg + δY . Conversely, if (L, ‖ ‖) aholomorphic line bundle with Hermitian metric, then its Green current is knownby the following theorem of Poincare-Lelong.

30

Proposition 6.1 For a meromorphic section s of L, −log‖s‖2 is a locally in-tegrable function X and gives a logarithmic Green function for for Y = div(s).Moreover the right hand side of

ddc(−log‖s‖2) + δY = c1(L, ‖ ‖)is the Chern form for the metric line bundle (L, ‖ ‖).

Existence of logarithmic Green currents are guaranteed in general, but littelis known about their concrete construction.

From now on, when X and Y are arithmetic quotients and Y is a divisor ofX, we show one way to construct the logarithmic Green current of Y .

6.2.2 The secondary spherical functions for affine symmetric pairs (G, H) ofrank 1

For our construction, we need a pair (G,H) of real reductive Lie groups satis-fying the follwoing ”axioms”:

(i) G, a connected real semisimple (algebraic) Lie group such that the quotientG/K by a maximal compact subgroup K is Hermitian symmetric (K = Gθ

with θ a Cartan involution);

(ii) H, a reductive subgroup of G, such that there exists an involution σ :G → G satisfying θσ = σθ and (Gσ)0 ⊂ H ⊂ Gσ. Moreover H\G is asemisimple symmetric space of real rank 1.

Then for the Lie algebras g = Lie(G), h = Lie(H) we have

g = gσ ∩ gθ ⊕ g−σ ∩ gθ ⊕ gσ ∩ g−θ ⊕ g−σ ∩ g−θ

and a maximal abelian subalgebra a in the last factor g−σ ∩ g−θ is of dimension1.

By the classification table of symmetric spaces, we have two cases:

(U− type) : g = su(n, 1), h = s(u(1)× u(n− 1, 1))(O− type) : g = so(n, 2), h = so(n− 1, 2).

We choose a generator Y0 ∈ a such that λ(Y0) = 1 for the short root λ in the rootsystem Ψ = Ψ(a, g). Set U = exp(gλ + g2λ) and 2ρ0 = tr(ad(Y0)|Lie(U)). Alsowe normalize the Casimir operator Ω such that it corresponds to the bilinearform

X,Y =1

B(Y0, Y0)B(X, Y ) (X, Y ∈ g).

Set A = exp(a) = at = exp(tY0)|t ∈ R, then G = HAK.Now we can consider a left H-invaraint and right K-invariant spherical func-

tion φ(1)s (g) ∈ C∞(H\G/K) satisfying

φ(1)s (g) ∗ Ω = (s2 − ρ2

0)φ(1)s (g) (s ∈ C).

This function generates class one principal series representation of G in C∞(H\G)by right translation under G, which has H-invariant. This ordinary sphericalfunction is not necessarily good one to define Poincare series. In place of this,we consider the secondary spherical functions φ(2)(g) ∈ C∞(G −H ·K) whichhas logarithmic singularities along H ·K.

31

Proposition 6.2 Let s ∈ C, Re(s) > ρ0. Then there exists the unique functionsatisfying the conditions (a)-(d) below:(a) φ

(2)s is C∞ on G−H ·K, and left H-invaraint and right K-invariant;

(b) φ(2)s satisfies the diferential equation:

φ(2)s ∗ Ω = (s2 − ρ2

0)φ(2)s on G−HK;

(c) For sufficiently small δ > 0,

φ(2)s (at)− log(t) is bounded in (0, δ);

(d) φ(2)s (at) is rapidly decreasing for t → +∞.

For our later purpose, it is better to introduce a vector-valued sphericalfunction ψs. Recall the Cartan decomposition and its complexification:

g = k⊕ p, gC = kC ⊕ p+ ⊕ p−.

Here the subpace p± is the ±i-eigenspaces with respect to the given complexstructure in the complexification p⊗C.

For a function F ∈ C∞(G/K) we can define the gradient ∇F = ∇+F +∇−F ∈ C∞(G) ⊗ pC. Then we have ∇−∇+F : G → p+ ⊗ p−, which is K-equivariant under the right K-action on G and the tensor product of the adjointrepresentation Adp± . The K-module p+ ⊗ p− is a direct sum of the trivialrepresentation and the other irreducible representation V11. We denote by prthe projection to V11 from p+ ⊗ p−

Definition 6.1 φs = pr · ∇−∇+φ(2)s .

6.2.3 Poincare series

We can now define Poincare series.

Definition 6.2 For s ∈ C with Re(s) > ρ0, set

Gs(z) =∑

γ∈(Γ∩H)\Γφ(2)

s (l, γ(z)) (g ∈ G)

andΨs(z) =

∑

γ∈(Γ∩H)\Γψ(2)

s (l, γ(z)) (g ∈ G).

Both converge as currents for Re(s) > ρ0 and real analytic except for on theimage of

(Γ ∩H)\H/(K ∩H) → Γ\G/K.

We can show

(i) a criterion for Gs ∈ Lp(Γ\G) in particular Gs ∈ L2(Γ\G),

(ii) the analytic continuation, when Gs ∈ L2(Γ\G), (this is a kind of functionalequation for Gs and G−s.

(iii) Gs(g) has a simple pole at s = ρ0 with residue

vol((H ∩ Γ)\H)vol(Γ\G)

12ρ0

.

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6.2.4 ∂∂-formula

Let d = ∂ + ∂ be the decompostion of the exterior derivative on G/K into theholomorphic part and the antiholomorphic part.

Theorem 6.1 We have

(a) ∂∂Gs + πδD0 = −√−12n

((c(g)s)2 − ρ20)Gs ∧ ωΓ\G/K + 4Ψs

(b) ∆Ψs = −((c(g)s)2 − ρ20)

(Ψs − π

√−14

δD0 +π√−14n

δD0 ∧ ωΓ\G/K

)

Here Ψs is the Poincare series of (1,1)-form, δD0 the current associated withthe divisor

(H ∩ Γ)\H/(H ∩K) → Γ\G/K,

c(g) a constant given by c(g) = 1 (U-type), = 2 (O-type), and ωΓ\G/K theKaehler form on Γ\G/K.

Theorem 6.2(i) The (1, 1)-type current Ψs is holomorphic at s = ρ0, and Ψρ0 is harmonicC∞-form of (1, 1)-type on Γ\G/K.　(ii) Put G = lims→ρ0(Gs − 1

s−ρ0Ress−ρ0Gs). Then

√−1∂∂G − πδD0 = κωΓ\G/K + 4√−1Ψρ0 .

(cf. [TsO01,Chapter 7, Theoem (7.6.1)] and its application).

Here is the meaning of the last formula. There is the subspace of square-integrable forms H2

(2)(Γ\G/K,C) in H2(Γ,C) = H2(Γ\G/K,C). There are twoirreducible unitary representation of G which contribute to the (1,1)-type Hodgecomponent of H2

(2)(Γ\G/K,C): one is the trivial representationC and the otheris a certain infinite dimensional representations, which we denote by π1,1. Bothrepresentations π have H1,1(g, K; H1,1) ∼= C. Hence

H1,1 = HomG(1, L2(Γ\G))⊕HomG(π1,1, L2(Γ\G)).

The first factor of the right hand side is generated by the Kaehler form κ. Theright hand side of the statement (ii) of the last theorem is compatible with thisdecomposition.

Postscript

We have no ability to see the future of our project. But let me comment aboutsome new directions in progress. For the spherical functions on G = Sp(2,R),we are interested in the relation between different type spherical functions. i.e.,they are different realization of the same irreducible unitary representation G.Therefore there should be an isomorphism of (g,K)-modules, and it is an in-teresting problem to find explicit realization of this intertwining isomorphism.About this direction, the author, Miki Hirano and Taku Ishii are obtaining somefomulae of ”confluence from Siegel-Whittaker functions to Whittaker functions”.

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This is, heuristically speaking, the application of the idea of ”contraction or de-formation of spherical subgroups”. This might be the first step toward theseamless formulae to encompass the various spherical functions.

Another direction is the joint work with Hirano to have explicit formula forthe radial part of Whittaker functions of the PJ principal series representationof Sp(3,R), whose Bernstein degree is the half of the order of the Weyl group,i.e., 24 = 1

2 (3!23). We have already some explicit formulae for the secondaryWhittaker functions, which are 24 independent power series solutions of theholonomic system consisting of the radial parts of the annihilators in U(g) ofthe corner K-type of PJ series.

In the above two attempts, the secondary spherical functions, which is theanalogue of Harish-Chandra’s hypergeometric series in his classical theory ofspherical functions on semisimple Lie groups, play a crucial role.

Yet another direction is to have Green currents of higher-codimensional mod-ular algebraic cycles. Here we have to consider slightly more general sphericalfunctions proposed by Masao Tsuzuki. But here also the ”secondary” functions,which have natural singularities inside G, is the seed of the whole construction.

At somewhere Jean-Pierre Serre said that Representation Theory and Alge-braic Geometry are the most difficult fields in mathematics. In the arithmetictheory of automorphic forms the One from which everything comes out is thediscrete subgroup Γ, which is the unit group of a non-commutaive arithmeticalgebra. The emanated objects are written in these two sophisticated languages.But if one wants to have arithmetic results, the direct application of the knownresults in these two fields is not enough. At least one has to have coin effectivecomputable results beyond the general framework.

This is really a demanding and time-consuming job. The only wise way tocope with this difficulty seems for one to have one’s own style or philosophy;not to follow the fashion of the time to waste time in vain.

References

[1] Borel, A.: Cohomologie des sous-groupes discrete et representations degroupses . Asterisque 32-33 (1976), 73–112

[2] Borel, A. and Wallach, N.: Continuous cohomology, discrete subgroups,and representations of reductive groups. Annals of Math. Studies No.94,Princeton Univesity Press. 1980; the 2-nd ed., Amer. Math. Soc. 2000.

[3] Dixmier, J.: Les C∗-algebres et leur reprentations, Gauthier-Villars, Paris,1964.

[4] Gon, Y.: Generalized Whittaker functions on SU(2, 2) with respect to Siegelparabolic subgroup, Memoir of Amer. Math. Soc. 155 (2002), no. 738,

[5] Harish-Chandra: Automorphic forms on semisimple Lie groups, LectureNotes in Math. 62, Springer, 1968.

[6] Hayata, T.: Some real Shintani functions over SU(2, 2). (Japanese) Re-search on automorphic forms and zeta functions. (in Japanese) (Kyoto,1997). Surikaisekikenkyusho Kokyuroku No. 1002 (1997), 160–168.

34

[7] Hecht, H. and Schmid, W.: A proof of Blattner’s conjecture. Invet. Math.31 (1975), 129–154.

[8] Hirai, T.: Explicit form of the characters of discrete series representationsof semisimple Lie groups. Harmonic analysis on homogeneous spaces (Proc.Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass.,1972), pp. 281–287. Amer. Math. Soc., Providence, R. I., 1973.

[9] Hirano, M.: Fourier-Jacobi type spherical functions for discrete series rep-resentations of Sp(2,R). Compositio Math. 128 (2001), 177–216.

[10] Hirano, M.: Fourier-Jacobi type spherical functions for PJ -principal seriesrepresentations of Sp(2,R). J. London Math. Soc. (2) 65 (2002), 524–546.

[11] Hori, A.: Andrianov’s L-functions associated to Siegel wave forms of degreetwo. Math. Ann. 303 (1995), 195–226.

[12] Hotta, R. and Parthasarathy, R.: Multiplicity formulae for discrete series.Invent. math. 26 (1974), 133–178.

[13] Ibukiyama, T. : On relations of dimensions of automorphic forms ofSp(2,R) and its compact twist Sp(2). I. Automorphic forms and numbertheory (Sendai, 1983), 7–30, Adv. Stud. Pure Math., 7, North-Holland,Amsterdam, 1985.

[14] Hashimoto, K., Ibukiyama, T.: On relations of dimensions of automorphicforms of Sp(2,R) and its compact twist Sp(2). II. Automorphic forms andnumber theory (Sendai, 1983), 31–102, Adv. Stud. Pure Math., 7, North-Holland, Amsterdam, 1985.

[15] Ibukiyama, T. and Ihara, Y.: On automorphic forms on the unitary sym-plectic group Sp(n) and SL2(R). Math. Ann. 278 (1987), 307–327.

[16] Ihara, Y.: On certain arithmetical Dirichlet series. J. Math. Soc. Japan 16(1964), 214–225.

[17] Iida, M.: Spherical Functions of the Principal series representations ofSp(2;R) as Hypergeometric Functions of C2-type. Publ. RIMS., Kyoto Uni-versity, 32 (1996), 689-727

[18] Ishii, T.: Siegel-Whittaker functions on Sp(2,R) for principal series repre-sentations. J. Math. Univ. Tokyo 9 (2002), 303–346.

[19] Ishikawa, Y.: The generalized Whittaker functions for SU(2, 1) and theFourier expansion of automorphic forms. J. Math. Sci. Univ. Tokyo 6(1999), no. 3, 477–526.

[20] Kobayashi, T.: Discrete decomposability of the restriction of Aq(λ) with re-spect to reductive subgroups and its applications. Invent. math. 117 (1994),181–205.

[21] Kobayashi, T.: Discrete decomposability of the restriction of Aq(λ), PartII. Micro-local analysis and asymptotic K-support. Annals of Math. 147(1998), 709–726.

35

[22] Kobayashi, T.: Discrete decomposability of the restriction of Aq(λ), PartIII. -Restriction of Harish-Chandra modules and associated varieties . In-vent. math. 131 (1998),229–256

[23] Kobayashi, T. and Oda, T.: A vanishing theorem for modular symbols onlocally symmetric spaces. Comment. Math. Helv. 73 (1998), 45–70.

[24] Kohnen, W. and Skoruppa, N.-P.:　 A certain Dirichlet series attached toSiegel modular forms of degree two.　 Invent. Math. 95 (1989), no. 3, 541–558.

[25] Kumaresan, S.: On the canonical k-types in the irreducible unitary g-modules with nonzero relative cohomology. Invent. Math. 59 (1980), 1–11.

[26] Langlands, R.: On the functional equations satisfied by Eisenstein series,Lecture Notes in Math., 544, Springer, 1976.

[27] Matsushima, Y.: On Betti numbers of compact, locally sysmmetric Rie-mannian manifolds. Osaka Math. J. 14 (1962), 1–20.

[28] Matsushima, Y.: On the first Betti number of compact quotient of higherdimensional symmetric spaces, Annals of Math. 75 (1962), 312–30.

[29] Miyazaki, T.: The Generalized Whittaker functions for Sp(2,R) and theGamma Factor of the Andrianov L-function, J. Fac. Sci. Univ. Tokyo 7(2000), 241–295.

[30] Miyazaki, T. and Oda, T.: Principal series Whittaker functions forSp(2;R). - Explocit formulae of differential equations - , Proc. of the 1993workshop, Automorphic forms and related topics, The Pyungsan Institutefor Math. Sci., 59–92.

[31] Miyazaki, T. and Oda, T.: Principal series Whittaker functions on Sp(2;R)II, Tohoku Math. J. (2) 50 (1998), 243–260., Errata Tohoku Math. J. (2)54 (2002), 161–162.

[32] Moriyama, T.: Spherical functions with respect to the semisimple symmetricpair (Sp(2,R),SL(2,R) × SL(2,R)). J. Math. Sci. Univ. Tokyo 6 (1999),no. 1, 127–179.

[33] Moriyama, T.: Spherical functions for the semisimple symmetry pair(Sp(2,R), SL(2,C)). Canad. J. Math. 54 (2002), no. 4, 828–865.

[34] Moriyama, T.: A remark on Whittaker functions on Sp(2,R). J. Math. Sci.Univ. Tokyo 9 (2002), 627–635.

[35] Moriyama, T.: On the spinor L-functions for generic cusp forms onGSp(2), To appear in Amer. Journ. of Math.

[36] Murase, A. and Sugano, T. : Shintani function and its application to auto-morphic L-functions for classical groups. I. The case of orthogonal groups.Math. Ann. 299 (1994), no. 1, 17–56.

[37] Narasimhan, M. S. and Okamoto, K.: An analogue of the Borel-Weil-Botttheorem for hermitian symmetric pair of non-compact type, Ann. of Math.,91 (1970), 486–511.

36

[38] Narita, H.: Fourier expansion of holomorphic Siegel modular forms withrespect to the minimal parabolic subgroup, Math. Z. 231 (1999), 557–588.

[39] Niwa, S. : On generalized Whittaker functions on Siegel’s upper half spaceof degree 2. Nagoya Math. J. 121 (1991), 171–184.

[40] Oda, T.: Hodge structures attached to geometric automorphic forms.Adv.Stud. in pure Math. 7 (1985), 223–276

[41] Oda, T.: An explicit integral representation of Whittaker functions onSp(2;R) for the large discrete series representations, Tohoku Math. J. 46(1994), 261-279

[42] Oda, T. and Schwermer, J.: Mixed Hodge structures and automorphic formsfor Siegel modular varieties of degree two. Math. Ann. 286 (1990), 481–509.

[43] Oda, T. and Tsuzuki, M. : Automorphic Green fucntions associated withthe secondary spherical functions . Publ. RIMS, Kyoto Univ. 39 (2003),451–533.

[44] Schmid, W.: On then realization of the discrete series of a semisimple Liegroup, Rice Univ. Stud. 56 (1970), 99-108

[45] Schmid, W.; On a conjecture of Langlamds, Ann. of Math., 93 (1971), 1–42.

[46] Schimid, W.: Some properties of square-interable represenations ofsemisimple Lie groups. Annals of Mathematics 102 (1975), 535-564.

[47] Schwwermer, J.: Eisenstein series and cohomology of arithmetic groups:the generic case. Invent. Math. 116 (1994), 481–511.

[48] Soule, C.: Lectures on Arakelov geometry. With the collaboration of D.Abramovich, J.-F. Burnol and J. Kramer. Cambridge Studies in AdvancedMathematics, 33. Cambridge University Press, Cambridge, 1992.

[49] Tsuzuki, M.: Real Shintani functions on U(n, 1). J. Math. Sci. Univ. Tokyo8 (2001), 609–688; II. Computation of zeta integrals. 8 (2001), 689–719; III.Construction of intertwining operators. 9 (2002), 165–215.

[50] Vogan, D. and Zuckerman, G.: Section 1: Unitary representations withnon-zero cohomology . Comp. Math.53 (1984), 51–90.

[51] Wong, H.-W.: Dolbeault cohomological realization of Zuckerman modulesassociated with finite rank representations. Journ. of Funct. Analy. 129(1995), 428–454

[52] Yamashita, H.: Embeddings of discrete series into induced representationsof semisimple Lie groups I, - general theory and the case of SU(2, 2) -,Japan J. Math. (N.S.) 16 (1990), 31–95.

[53] Zucker, S.: L2-cohomology of Shimura varieties. Automprphic forms,Shimura varieties, and L-functions, Vol. II (Ann Arbor, MI, 1988), 377–391,Perspect. Math., 11, Academic Press, Boston, MA, 1990.

[54] Zuckerman, G.: Continuous cohomology and unitary representations of realreductive groups. Ann. of Math. 107 (1978), 495–516.

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Graduate School of Mathematical SciencesThe University of TokyoKomaba 3-8-1, Meguro Ward, Tokyo 153-8914JAPAN

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